If f (x) = 3x-5 find f (x) -f (a) / x-a

Question

What is the domain , range and function notation 

Answer 1

f(x) is already defined as it is.

f(a) is f(x) evaluate when x=a.

 

 

       f(x) - f(a)

   ____________  =

         x - a

 

 

  (3x - 5) - (3a - 5)

 _________________ =

        x - a

 

 

 

      3x - 3a

   _________ =

       x - a

 

 

 

Domain is the values of x in which function is defined.  Range is the values of y in which the function is defined.  Since we cannot have a zero in the denominator, we set the denominator equal to zero.

 

x - a = 0

 

x = a

 

 

Domain is all values of x except for a.

 

(-∞, a)∪(a, ∞).

 

 

To find the range, we take the limit of the function as x approaches infinity.  This is also how we determine the horizontal asymptote.

 

lim           3x - 3a

x-->∞     _______

                x - a

 

 

lim          3 (x - a)

x-->∞     _______

                  x - a

 

 

lim       3

x-->∞

 

 

The range is (-∞, 3)∪(3, ∞).

Answer 2

Let's deal with function notation first - that means simplifying this function.

 

f(x) - f(a)/x-a   

 

Using f(x) = 3x-5, the above would look like this:

 

[(3x-5) - (3a-5)]/x-a

 

The top turns into:

(3x - 5 - 3a + 5)/x-a

 

then:

(3x-3a)/x-a = 3(x-a)/x-a = 3

 

The domain is anything that limits the x value - in this case, it would be limited by the denominator.  The denominator can't equal 0.  Since we have x-a on the bottom, then x cannot equal a because if it did, then the bottom would be zero.  So the domain is all real numbers except x cannot equal a.

 

The range is the set of resulting y values.  What you see from the above equation is that no matter what x or a we put in, y will always equal 3, so that is your range: y=3.

Answer 3

f(x) = 3x-5

 

[f(x)-f(a)]/(x-a) = [3x-5 - (3a-5)]/(x-a)

 

                      = [3x-5-3a+5]/(x-a)

 

                      = (3x-3a)/(x-a)

 

                      = 3(x-a)/(x-a) = 3

 

The graph of f is a line with slope 3 and y-intercept -5.

 

Domain = (-∞, ∞)     Range = (-∞, ∞)